On the Classification of Vaisman Manifolds with Vanishing First Basic Chern Class and Large First Betti Number

TL;DR

This paper classifies certain Vaisman manifolds with specific topological and geometric properties, showing they are diffeomorphic to Kodaira-Thurston manifolds and analyzing their complex structures.

Contribution

It establishes a classification of Vaisman manifolds with large first Betti number and vanishing first basic Chern class, highlighting their diffeomorphism to Kodaira-Thurston manifolds and invariance properties.

Findings

Vaisman manifolds with specified conditions are diffeomorphic to Kodaira-Thurston manifolds.

The complex structure on these manifolds is left-invariant under certain conditions.

If basic harmonic 1-forms have constant length, the manifold's complex structure is standard.

Abstract

We show that every Vaisman manifold with large first Betti number and vanishing first basic Chern class is diffeomorphic to a Kodaira-Thurston manifold. Furthermore, its complex structure is left-invariant, the characteristic foliation is regular, and the associated fibration is given by the Albanese map. Under the additional assumption that the LCK rank is $1$, the Vaisman structure is also left-invariant. We further prove that if all basic harmonic $1$-forms have constant length, then the Vaisman manifold with large first Betti number is diffeomorphic to a Kodaira-Thurston manifold and its complex structure is the standard complex structure. Finally, we discuss the relationship of this condition with transverse geometric formality in this setting.